Schreier graphs of the Basilica group

TL;DR

This paper classifies the limit Schreier graphs of the Basilica group acting on a binary tree, based on the infinite binary sequences, providing a comprehensive understanding of their structure.

Contribution

It offers a complete classification of the limit graphs associated with the Basilica group, linking their structure to infinite binary sequences.

Findings

Classification of limit graphs up to isomorphism

Connection between binary sequences and graph structures

Complete description of Schreier graph limits for the Basilica group

Abstract

With any self-similar action of a finitely generated group $G$ of automorphisms of a regular rooted tree $T$ can be naturally associated an infinite sequence of finite graphs $\{\Gamma_n\}_{n\geq 1}$, where $\Gamma_n$ is the Schreier graph of the action of $G$ on the $n$-th level of $T$. Moreover, the action of $G$ on $\partial T$ gives rise to orbital Schreier graphs $\Gamma_{\xi}$, $\xi\in \partial T$. Denoting by $\xi_n$ the prefix of length $n$ of the infinite ray $\xi$, the rooted graph $(\Gamma_{\xi},\xi)$ is then the limit of the sequence of finite rooted graphs $\{(\Gamma_n,\xi_n)\}_{n\geq 1}$ in the sense of pointed Gromov-Hausdorff convergence. In this paper, we give a complete classification (up to isomorphism) of the limit graphs $(\Gamma_{\xi},\xi)$ associated with the Basilica group acting on the binary tree, in terms of the infinite binary sequence $\xi$.

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